Conditional κ = 7.93392809291691·10⁻⁴: #346's w3 bit word with a per-point p = 10 complex supplier (+0.1025 % over #346)

[chafreaky]

Conditional κ = 7.93392809291691·10⁻⁴: #346's w3 bit word with a per-point p = 10 complex supplier (+0.1025 % over #346)

This PR adds one self-contained package, research/w3-p10b-pointwise-complex/, with its own workflow. No existing file changes except one .gitattributes line.

κ = 793392809291691/10¹⁸ ≈ 7.93392809291691·10⁻⁴. That is +812715921745/10¹⁸ (+0.1025 %) over #346 (396290046684973/(5·10¹⁷) ≈ 7.92580093369946·10⁻⁴). The complex side binds. #346's bit coarse saving, c ≈ 8.01726·10⁻⁴, is 0.97 % above the new leaf cap.

Change and mathematical scope

Bit side: #346's, unchanged. #346 (LJH-217) applied #310's w3 Design T and twin condensation to #325's p10b word. Its final records, hash list, κ receipt, LICENSE and NOTICE are vendored in bit/pr346/, pinned by git blob at 11b4de5. verify.py does the following:

• decodes the records with its own reader;
• recounts the paid children (MOVE/COPY ranks);
• requires κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346's normalized histogram to be exactly 60·H + 12·2v·(e₃₈ + e₁₉ + e₄₂ + e₄);
• certifies c = 32069035133443/(4·10¹⁶) with both rational engines and the 10⁻¹⁶ fallback, which equals κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346's c.

The stock W = 125,712 is taken from #346's receipt. #346's own bit-word checks (Design T verifier, #266's checkers, primes, banks) are its claim and are not re-run.

Complex side: a stronger p = 10 program of the same recipe. It uses #327's chain (pinned at 40ed045):

• A p = 10 complex supplier: five-stage complex saving 7795/10⁷ (centre-sharing) and 7737/10⁷ (#304 recipe), both kernel-checked in Lean (no new κ; raises the complex cap above 7727) #327's triple module tmod_t2_p10;
• Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word) #325's all-but-one module qmod_p10 (Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's);
• a separate pair module for each point i, shared by both bits (modules/pointwise/, six distinct files).

The pair modules come from a greedy climb on the exact compile proxy, starting from #325's pm_p10. A per-invocation module spec (spec/c2.json) names them. spec_graph.finish_spec is #327's finish_centre46 with the module looked up per invocation, and aligned_word_spec.py is #327's aligned_word_p.py with a --spec branch. On a one-module-per-kind spec, two anchors check that they reproduce #327's graph and seven word files byte for byte.

complex program (five-stage, h = 20, m = 100)	R	W	N	a	coarse b (10⁻¹⁶ fallback)	Lean
#327 centre-mw	6,455	10,295	165,940	7795/10⁷	779597612622781/10¹⁸	1 − 7795973/10¹⁰
#346's supplier / shared #325 modules	6,265	10,105	162,140	7932/10⁷	793208776027046/10¹⁸	1 − 7932084/10¹⁰ (our run, companion)
this PR: pointwise pair modules	6,253	10,093	161,900	7940/10⁷	794022782431403/10¹⁸	pending

κ. We use #315's math_check outer assembly on the complex-bound path, as in our #329. The bit saving is capped one 10⁻¹⁸ grid point below (1 − 10⁻⁹)·b, followed by three bootstrap steps, η = 10⁻¹² and #315's outer.assembly with 47 strict constraints and the finite bridge. The adjacent κ is rejected, and so is the leaf cap itself (leaf_saving_above_bit). Run on #346's own b, this computation reproduces #346's κ exactly. Run on ours, it gives 793392809291691/10¹⁸.

Status of the 7940/10⁷ program. It passes Sussman's gx.check1 (exact scalar identity) and his gxcore mirror of the two Lean checks, and both reject a flipped sign. It also passes every Python check of #327's chain:

• paircheck with 5 mutation controls;
• A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's complete complex checker;
• the exact flow lift;
• Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's contract, 13/13;
• byte-for-byte regeneration.

Its Lean run is pending, because the Lean host needs an interactive re-authentication. The companion program, which uses one shared #325 pair module and has the same b as #346's supplier, went through the same generator path. Its Lean run passed (wht_main_block_B2Gcu32x at 1 − 7932084/10¹⁰, Compare.lean PASS, axioms propext/Classical.choice/Quot.sound; console output in lean/companion-7932-evidence.txt). The per-point program differs from it only in module files.

Retained hypotheses.

• All inherited interfaces of Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315/Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word) #325/κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346 are retained.
• κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346's open bit-side obligations are retained: the w3 downstream chain is not ported to p = 10, the edited-word scalar/prime/finite stages are not re-run, and the p = 11 additive row constants of the complex guard are kept.
• Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's portable complex checks (ported by κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346) are not run on this program.
• The B₂ ledger at h = 20 and the 10⁻¹⁶ fallback envelope are assumed.

This is a conditional finite construction, not a theorem.

Validation

python -m pip install numpy==2.3.5 scipy==1.17.0
python3 -B research/w3-p10b-pointwise-complex/verify.py --quick   # ~30 s
python3 -B research/w3-p10b-pointwise-complex/verify.py           # ~11 min, regenerates the complex program

Local result (macOS, Python 3.14.8, numpy 2.3.5, scipy 1.17.0), branch w3-p10b-pointwise-complex at 550fb36:

$ verify.py --quick
[  22.3s] complex p10-b325-c2: five-stage word at h = 20 (m = 100, W = 10093, deficit 2040) certified at a = 7940/10^7 with and without the 10^-16 fallback, 7941/10^7 rejected; coarse b = 794022782431403/10^18 with the fallback, 794022785498833/10^18 without (both engines)
[  22.3s] PR #346 reproduced: complex b = 396604388013523/500000000000000000 binds, kappa = 396290046684973/500000000000000000 = 7.925800933699460e-04
[  22.4s] this package: complex b = 794022782431403/10^18 binds (bit c = 801725878336075/10^18 above the leaf cap); kappa = 793392809291691/1000000000000000000 = 7.933928092916910e-04, +0.1025% over PR #346; 47 strict constraints, adjacent kappa and the leaf cap itself rejected
[  22.4s] PASS certificate/expected.json reproduced (without the regeneration entry): conditional kappa = 793392809291691/1000000000000000000 ≈ 7.933928092916910e-04 (complex-bound; PR #346: 396290046684973/500000000000000000, +0.1025%)
$ verify.py
[  69.9s] anchor (a): aligned_word_p.py at p = 11 reproduces the 7 output files of source_aligned_local_v4.py byte for byte
[ 166.1s] anchor (b): PR #194's word through this chain is PR #256's certificate (sha256 27398a0cfcbe...) with Sussman's #193 histogram
[ 166.2s] anchor (d): PR #315's Graph.finish patch builds centre46.py's graph on PR #325's modules; anchor (e): spec_graph.finish_spec on spec/uniform.json builds the same graph (every field)
[ 185.7s] anchor (f): on the companion layer, aligned_word_spec.py --spec spec/uniform.json writes the 7 files of aligned_word_p.py --centre46 byte for byte
[ 280.5s] regenerated program = committed program byte for byte (uncompressed canonical JSON, sha256 ff2d7307b09a...)
[ 280.5s] PASS certificate/expected.json reproduced: conditional kappa = 793392809291691/1000000000000000000 ≈ 7.933928092916910e-04 (complex-bound; PR #346: 396290046684973/500000000000000000, +0.1025%)

The workflow .github/workflows/w3-p10b-pointwise-complex.yml runs --quick and the full run on Ubuntu with Python 3.12 and 3.14, then git diff --exit-code.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0. The full lineage is in the package NOTICE.

• LJH-217 / Louis Harrison (κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346): the w3 bit word on Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word) #325's p10b word (vendored), and the complex-bound composition whose κ this package reproduces.
• utcorvusvolat-dotcom (Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310): w3 Design T and twin condensation.
• Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266: the official cohort checkers.
• DreamingOfClouds:
  • Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word) #325's p10b word, the pm_p10 pair module (the climb's start and the spec's default) and the all-but-one module;
  • Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's p = 10 package and outer-assembly convention;
  • A p = 10 complex supplier: five-stage complex saving 7795/10⁷ (centre-sharing) and 7737/10⁷ (#304 recipe), both kernel-checked in Lean (no new κ; raises the complex cap above 7727) #327's p = 10 complex recipe, chain, triple module and verify.py, which this package generalises.
• Jacob Sussman: gcert/1, gx.py/gxcore.py, B₂, and the Lean generator and comparator.
• Complex lineage:
  • our A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word, complex saving 7.0991·10⁻⁴ (+1.28% over #194; no new κ, bit binds) #233 pairing, A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 descent and checker, The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256 emitter and The complex suppliers transfer to the exact DFT: θ = 1 − 7009/10⁷ (7.009·10⁻⁴) for the exact discrete Fourier transform (no new κ) #225 bounds;
  • icekylinx's Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144/Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184;
  • Avi Eisenberg's Source-assisted complex word on PR168 v4 modules: conditional κ = 6.626307e-4 #191/Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193/Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194/Compose the v4 source-assisted complex word with the PR200 bit supplier: conditional κ = 6.768823e-4 #202;
  • eumemic's Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 modules, source527 engines and outer.py;
  • DaysSky's Extended carrier matching on #161's complex word: conditional κ = 5.901881e-4 #162, hcg890's Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234 and SovereignSteak's Compose terminal query retention into the five-stage complex supplier #250;
  • the Compose the v4 source-assisted complex word with the PR200 bit supplier: conditional κ = 6.768823e-4 #202 baseline lineage.

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Per-point complex modules paired with the w3 bit word.

Review and limits: Full complex regeneration and exact pricing; the bit side partly reuses source-bound receipts and leaves native obligations explicit. No new Lean build is inferred.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 2b8b3d87870f68e646d295bb2d32c6dd918dee9d. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.